Digital Object Identifier (DOI) 10.1007/s00220-008-0552-z
Mathematical
Physics
The Pauli Principle Revisited
Murat Altunbulak, Alexander Klyachko
Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey. E-mail: murata@fen.bilkent.edu.tr; klyachko@fen.bilkent.edu.tr
Received: 22 November 2006 / Accepted: 25 February 2008 Published online: 4 July 2008 – © Springer-Verlag 2008
Abstract: By the Pauli exclusion principle, no quantum state can be occupied by more
than one electron. One can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1. Shortly after its discovery, the Pauli principle was replaced by anti-symmetry of the multi-electron wave function. In this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix, that goes far beyond the original Pauli principle. Our approach uses Berenstein and Sjamaar’s theorem on the restriction of an adjoint orbit onto a subgroup, and allows us to treat any type of permutational symmetry.
Electronic Supplementary Material: The online version of this article
(doi:10.1007/s00220-008-0552-z) contains supplementary material, which is available to authorized users.
Contents
1. Introduction . . . 287
2. A Review of the Berenstein-Sjamaar Paper . . . 290
3. One Pointν-Representability . . . 293
4. Beyond the Basic Constraints . . . 304
5. Connection with Representation Theory. . . 316
6. Analysis of Some Small Systems . . . 317
1. Introduction
The Pauli exclusion principle, discovered in 1925, claims that no quantum state can be occupied by more than one electron. In terms of the electron density matrix1 ρ 1 There is no agreement on a proper normalization of the one-electron matrix. To avoid confusion we call
it electron density matrix for Dirac’s normalization to the number of particles Trρ = N, and reserve the term reduced state for the probability normalization Trρ = 1.
this amounts to the inequalityψ|ρ|ψ ≤ 1, that bounds its eigenvalues by one. The following year Heisenberg and Dirac replaced the Pauli principle by skew symmetry of a multi-electron wave function [11, Ch. 4].
The subject of this study is the impact of this replacement on the electron density matrix. The latter determines the light scattering and therefore quite literally represents a visible state of the electron system. The impact goes far beyond the original Pauli principle. As an example, consider a three electron system ∧3H6 with one-electron
spaceH6of dimension 6. Then the spectrumλ of the electron density matrix, arranged
in non-increasing order, is bounded by the following (in)equalities discovered by Borland and Dennis [3]:
λ1+λ6= λ2+λ5= λ3+λ4= 1, λ4≤ λ5+λ6. (1)
The authors established the sufficiency of these constraints and refer for a complete proof to M.B. Ruskai and R.L. Kingsley.2It worth reading their comment:
We have no apology for consideration of such a special case. The general N -representability problem is so difficult and yet so fundamental for many branches of science that each concrete result is useful in shedding light on the nature of general solution.
In spite of some bogus claims [29], refuted in [32], this result had stood for more than three decades as the only known solution of the N -representability problem beyond two electrons∧2Hr and two holes∧r−2Hr. For the latter systems the problem is easy and
the constraints amount to double degeneracy of the spectrum, starting from the head λ2i−1 = λ2i for two electrons and from the tailλr−2i = λr−2i−1 for two holes [5],
where we setλi = 0 for i > r, and λi = 1 for i < 1.
Here we solve this longstanding problem. The content of the paper is as follows. In Sect.2 we recast the Berenstein-Sjamaar theorem [1, Thm 3.2.1] into a usable form (Theorem1). This provides a theoretical basis for our study.
We start Sect. 3 by a variation of the above problem, called ν-representability, that takes into account both spin and orbital occupation numbers. Mathematically this amounts to replacing the exterior power∧NH by a representation Hνdefined by a Young diagramν of order N. Theorem2gives a formal solution of theν-representability pro-blem. We derive from it the majorization inequalityλ ν, that plays the rôle of the Pauli principle. This inequality is necessary and sufficient forλ to be occupation numbers of an unspecified mixed state (Theorem3). Theorem4deals with a class of systems where the majorization inequality alone provides a criterion for pureν-representability. This includes the so-called closed shell, meaning a system of electrons of total spin zero. The corresponding Young diagramν consists of two columns of equal length. For this system all constraints on the occupation numbers are given by the Pauli type inequality λ ≤ 2. Next in Theorem5we calculate the topological coefficients cvw(a) that governed the constraints on the occupation numbers in Theorem2. This gives it the full strength we need in the next section.
Section4starts with analysis of pureν-representability for a toy example of two-row diagrams, that allows us to illustrate the basic technique (Theorem6). These are excep-tional systems where the constraints on the occupation numbers are given by a finite set 2 Recently M.B. Ruskai published the proof [33] derived from known constraints on the spectra of Hermitian
matrices A, B, and C = A + B. Conceptually the N-representability problem is close to the Hermitian spectral problem [15,16], but a direct connection between them, beyond sporadic coincidences, is unlikely. R.L. Kingsley’s independent solution apparently has never been published.
of inequalities independent of the rank. Then we return to the original N -representability problem, that appears to be the most difficult one. For example, in contrast to Theorem6, no finite system of inequalities can describe N -representability for a fixed N > 1 and arbitrary big rank (Corollary 3to Proposition5). This forces us to restrict either the rank, as we do in the last section, or the type of the inequalities. Here we focus on the inequalities with 0/1 coefficients. It turns out that under some natural conditions such an inequality should be either of the form
λi1+λi2+· · · + λiN−1 ≤ N − 2, (2)
withk(ik− k) = r − N + 1, or of the form
λi1+λi2+· · · + λip ≤ N − 1, (3)
with p ≥ N andk(ik− k) =
p N
. We call them Grassmann inequalities of the first and second kind respectively. A surprising result is that these inequalities actually hold true with very few exceptions (Theorems7and8).
In the simplest case N = 3 we get from (2) inequalities λk+1+λr−k≤ 1, 0 ≤ k < (r − 1)/2
that hold for any even rank r ≥ 6. This constraint prohibits more than one electron to occupy two symmetric orbitals and supersedes the original Pauli principle. For r = 6, due to the normalizationiλi = 3, the inequalities degenerate into Borland-Dennis
equalities (1). For odd rank the first inequality k= 0 should be either skipped or replaced by the weaker oneλ1+λr ≤ 1 +r−12 .
We treat Grassmann inequalities of the second kind (3) only for lowest levels
p= N, N + 1. For N = 3 and p = N + 1 they amount to four inequalities:
λ2+λ3+λ4+λ5≤ 2, λ1+λ3+λ4+λ6≤ 2,
λ1+λ2+λ5+λ6≤ 2, λ1+λ2+λ4+λ7≤ 2, (4)
that hold for arbitrary rank r and give all the constraints for r ≤ 7. For r = 6 they turn into Borland-Dennis conditions (1).
In Sect.5we briefly discuss a connection of theν-representability with representation theory, that provides information complementary to Theorem2. A combination of the two approaches leads to an algorithm for solution of the problem for any fixed rank. The algorithm, along with other tools, has been used in calculations reported in the last Sect.6. Eventually this led to a complete solution of the N -representability problem for rank r ≤ 10. However, we provide a rigorous justification only for r ≤ 8. We also give an example of constraints on the spin and orbital occupation numbers for a system of three electrons of total spin 1/2.
The first sections may be mathematically more demanding than the rest of the paper. We recommend books [7–9] as general references on Schubert calculus, Lie algebra, and representation theory.
The theoretical results of the paper belong to the second author. They were often inspired by calculations, that at this stage couldn’t be accomplished by a computer without intelligent human assistance and insight.
2. A Review of the Berenstein-Sjamaar Paper
Let M be a compact connected Lie group with the Lie algebra m and its dual coadjoint representation m∗. For coadjoint orbitO ⊂ m∗ of group M and a Cartan subalgebra
t ⊂ m consider the composition : O → m∗ → t∗known as the moment map. By
Kostant’s theorem its image is a convex polytope spanned by the W -orbit of some weight µ ∈ t∗which can be taken from a fixed positive Weyl chamber t∗
+. Here W= N(t)/Z(t)
is the Weyl group of M. This gives a parameterization of the coadjoint orbitsOµby the dominant weightsµ ∈ t∗+.
Example 1. In this paper we will mostly deal with the unitary group U(n) whose Lie algebra u(n) consists of all Hermitian3n× n matrices. Let us identify u(n) with its dual via the invariant trace form(A, B) = Tr(AB). Then the (co)adjoint orbit Oµconsists of all Hermitian matrices A of spectrumµ : µ1 ≥ µ2 ≥ · · · ≥ µnand the moment
map : Oµ → t is given by orthogonal projection into the Cartan subalgebra of diagonal matrices t. Kostant’s theorem in this case amounts to Horn’s observation that the diagonal entries of Hermitian matrices of spectrumµ form a convex polytope with verticeswµ obtained from µ by permutations of the coordinates µi. This is equivalent
to the majorization inequalities
d1 ≤ µ1,
d1+ d2 ≤ µ1+µ2,
d1+ d2+ d3 ≤ µ1+µ2+µ3, (5)
· · · ·
d1+ d2+· · · + dn = µ1+µ2+· · · + µn
for the diagonal entries d : d1 ≥ d2 ≥ · · · ≥ dnof matrix A. We will use for them a
shortcut d µ.
Consider now an immersion f : L → M of another compact Lie group L and the induced morphisms f∗ : l → m and f∗ : m∗ → l∗ of the Lie algebras and their duals. In the paper [1] Berenstein and Sjamaar found a decomposition of the projection f∗(Oµ) ⊂ l∗of an M-orbitOµ⊂ m∗into L-orbitsOλ⊂ f∗(Oµ). Here we paraphrase their main result in a form suitable for the intended applications.
Fix Cartan subalgebras tL → tM, and for every test spectrum a∈ tL consider the
inclusion of the adjoint orbits of groups L and M,
ϕa: Oa→ Of∗(a), (6)
through a and f∗(a) respectively. Topologically the orbits are (generalized) flag varieties. They carry a hidden complex structure coming from the representation
Oa= L/ZL(a) = LC/Pa, (7)
where Pa⊂ LCis a parabolic subgroup of the complexified group LCwhose Lie algebra
pais spanned by tL and the root vectors Xα such thatα, a ≥ 0. One can say this in
another way:
Pa= {g ∈ LC| lim
t→−∞e
tage−taexists},
which makes it clear that f : Pa→ Pf∗(a).
3 Hereafter we treatu(n) as the algebra of Hermitian, rather than skew-Hermitian, operators at the expense
We will use the parabolic subgroups to construct canonical bases in cohomologies H∗(Oa) and H∗(Of∗(a)). Let TL ⊂ B ⊂ Pabe a Borel subgroup containing a maximal
torus TL with Lie algebra tL. The flag varietyOa = LC/Pasplits into a disjoint union
of Schubert cells BvPa/Pa, parameterized by the left cosets v ∈ WL/WZL(a) or in
practice by representatives of minimal length = (v) in these cosets. We actually prefer to deal with shifted cells v−1BvPa/Pa = BvPa/Pa depending on the Borel
subgroups Bv⊃ TLmodulo conjugation by the Weyl group of the centralizer W(ZL(a)).
The closure of BvPa/Pais known as the Schubert variety, and its cohomology class
σv ∈ H2 (v)(Oa) is called the Schubert cocycle. These cocycles form the canonical
basis of the cohomology ring H∗(Oa).
Inclusion (6) induces a morphism of the cohomologies ϕ∗
a : H∗(Of∗(a)) → H∗(Oa), (8)
given in the canonical bases by the coefficients cvw(a) of the decomposition ϕ∗
a : σw →
v
cvw(a)σv. (9)
They play a crucial rôle in the next theorem. We extend them by zeros if eitherv ∈ WLor
w ∈ WM is not the minimal representative of a coset in WL/WZL(a)or WM/WZM( f∗(a))
respectively.
Theorem 1. In the above notations the inclusion Oλ ⊂ f∗(Oµ) is equivalent to the
following system of linear inequalities
λ, va ≤ µ, w f∗(a) (a, v, w)
for all a∈ tL, v ∈ WL, w ∈ WM such that cvw(a) = 0.
Proof. This is not the way Berenstein and Sjamaar stated their result. Instead, for some generic a0∈ tLthey fix positive Weyl chambers t+L a0and t+M f∗(a0) and use them
to define Schubert cocyclesσv ∈ H∗(Oa) and σw ∈ H∗(Of∗(a)) for all other a ∈ t+L.
Hence their Schubert cocyclesσware canonical in the above sense iff f∗(a) and f∗(a0)
are in the same Weyl chamber. The set of such a ∈ t+L form a convex polyhedral cone called the principle cubicle. It is determined by a0, and different choices of a0produce
a polyhedral decomposition of the positive Weyl chamber t+L into cubicles.
For every cubicle Berenstein and Sjamaar gave a system of linear constraints on the dominant weights λ, µ, so that all together they provide a criterion for the inclusion Oλ⊂ f∗(Oµ). For the principal cubicle the constraints are simplest and are as follows [1, Thm 3.2.1]:
v−1λ ∈ f∗(w−1µ − C), for cwv(a0) = 0, (10)
whereC is a cone spanned by the positive roots in t∗M. Note that f∗(C) is the cone dual to the principal cubicle and therefore the above condition can be recast into the inequalities
v−1λ, a ≤ f∗(w−1µ), a ⇐⇒ λ, va ≤ µ, w f∗(a), (11) that hold for all a from the principle cubicle provided that cvw(a0) = 0. The coefficients
cvw(a) are actually constant inside the cubicle, and therefore the last condition can be changed to cvw(a) = 0. Thus we arrive at the inequalities (a, v, w) for the principle
one. They are equivalent to the remaining more complicated inequalities in [1, Thm 3.2.1], but look different since Berenstein and Sjamaar use other non-canonical Schubert cocycles.
Example 2. Quantum marginal problem [17]. Let’s illustrate the above theorem with immersion of unitary groups
f : U(HA) × U(HB) → U(HA B), gA× gB → gA⊗ gB,
whereHA B= HA⊗ HB. As we have seen in Example1the coadjoint orbit of U(HA B)
consists of the isospectral Hermitian operatorsρA B : HA B understood here as mixed
states. The projection
f∗(ρA B) = ρA⊗ 1 + 1 ⊗ ρB
amounts to reduced operatorsρA: HAandρB : HBimplicitly defined by the equations
TrHA(ρAXA) = TrHA B(ρA BXA), TrHB(ρAXB) = TrHA B(ρA BXB) (12)
for all Hermitian operators XA : HA and XB : HB. This means thatρA,ρB are just
the visible states of the subsystems HA,HB. In this setting Theorem1 tells us that
all constraints on the decreasing spectra λA B = Spec(ρA B), λA = Spec(ρA), and λB = Spec(ρB) are given by the inequalities
i aiλuA(i)+ j bjλBv( j)≤ k (a + b)↓kλ A B w(k), (13)
for all test spectra a : a1 ≥ a2 ≥ · · · ≥ an, b: b1≥ b2 ≥ · · · ≥ bm from the Cartan
subalgebras tA, tB and permutations u, v, w such that cuwv(a, b) = 0. Here (a + b)↓
denotes the sequence ai + bj arranged in decreasing order. The order determines the
canonical Weyl chamber containing f∗(a, b). The pairs (a, b) with fixed order of terms ai+ bj in(a + b)↓form a cubicle.
The adjoint orbit Oa ⊂ u(HA) is a classical flag variety understood as the set of
Hermitian operators XA : HAof spectrum a = Spec XA. Denote it byFa(HA). Then
the morphism (6) is given by the equation
ϕab: Fa(HA) × Fb(HB) → Fa+b(HA B), (XA, XB) → XA⊗ 1 + 1 ⊗ XB, (14)
and the coefficients cuwv(a, b) are determined by the induced morphism of the cohomo-logies
ϕ∗ab: H∗(Fa+b(HA B)) → H∗(Fa(HA)) ⊗ H∗(Fb(HB))
σw →
u,v
cuwv(a, b) · σu⊗ σv. (15)
One can find the details of their calculation in [17]. Note that cwuv(a, b) = 1 for identical permutations u, v, w. Hence we get for free the following basic inequality:
i aiλiA+ j bjλBj ≤ k (a + b)↓kλ A B k , (16)
3. One Pointν-Representability
In this section we apply the above results to the morphism f : U(H) → U(Hν) given by an irreducible representationHν of group U(H) with a Young diagram ν of order
N = |ν|. For a column diagram we return to the N-fermion system ∧NH, while a
row diagram corresponds to the N -boson space SNH. However, the main reason to consider the general para-statistical representationsHν is not a uniform treatment of fermions and bosons, but taking into account spin. Observe that the state space of a single particle with spin splits into the tensor productH = Hr⊗ Hs of the orbitalHrand the
spinHs degrees of freedom. The total N -fermion space decomposes into spin-orbital
components as follows [35]: ∧N(H r ⊗ Hs) = |ν|=N Hrν⊗ Hν t s , (17)
where νt stands for the transpose diagram. In many physical systems, like electrons in an atom or a molecule, the total spin is a well defined quantity that singles out a specific component of this decomposition. Theorem1applied to the component gives all constraints on the possible spin and orbital occupation numbers, see the details in n◦3.1.1below.
3.1. Physical interpretation. Let’s now relate Theorem1to the N -representability pro-blem and its ramifications indicated above. We’ll refer to the latter as theν-representability problem.
It is instructive to think about X ∈ u(H) as an observable and treat ρ ∈ u(H)∗as a mixed state with the duality pairing given by the expectation value of X in stateρ,
X, ρ = TrHXρ (18)
(forget for a while about the positivityρ ≥ 0 and normalization Tr ρ = 1).
We want to elucidate the physical meaning of the projection f∗: u(Hν)∗→ u(H)∗ uniquely determined by the equation
f∗(X), ρν = X, f∗(ρν), X ∈ u(H), ρν ∈ u(Hν)∗. In the above setting (18) it reads as follows:
TrHν(Xρν) = TrH(X f∗(ρν)), ∀X ∈ u(H). (19) A good point to start with is Schur’s duality between irreducible representations of the unitary U(H) and the symmetric SNgroups,
H⊗N =
|ν|=N
Hν⊗ Sν. (20)
The latter group acts onH⊗N by permutations of the tensor factors, and its irreducible representationsSν show up in the right-hand side. One can treatH⊗N as a state space of N -particles, and for identical particles all physical quantities should commute with SN. Looking into the right-hand side of (20) we see that such quantities are linear
combinations of operatorsρν⊗ 1 acting in the component Hν⊗ Sν and equal to zero elsewhere. In the case of a genuine mixed stateρν, i.e. a nonnegative operator of trace
1, one can treat(ρν⊗ 1)/ dim Sνas a mixed state of N identical particles obeying some para-statistics of typeν. Let ρi : H be its ithreduced state. Sinceρν⊗ 1 commutes with
SN, the reduced stateρ = ρi is actually independent of i . However, occasionally we
retain the index i just to indicate the tensor component where it operates.
Proposition 1. In the above notations
f∗(ρν) = Nρ. (21)
Proof. We have to check that (21) fits Eq. (19): TrHν(Xρν) = TrHν⊗Sν Xρ ν⊗ 1 dimSν = TrH⊗N X ρν⊗ 1 dimSν = i TrHXiρi = N TrHXρ,
where Xi is a copy of X acting in the ithcomponent ofH⊗N, so that
TrH⊗N Xiρ
ν ⊗ 1
dimSν = TrHXiρi by definition (12) of the reduced state.
A generalν-representability problem concerns the relationship between the spectrum µ of a mixed state ρνand spectrumλ of its particle density matrix Nρ. The latter spectrum is known as the occupation numbers4of the system in stateρν. Formally the constraints on the spectra are given by Theorem1.
Remark 1. The above construction allows for a given mixed stateρνto define the higher order reduced matrices. Their characterization would have almost unlimited applica-tions. Indeed, behavior of most systems of physical interest is governed by two-particle interaction. As a result, the energy of a state becomes a linear functional of its two-point reduced matrix. To minimize the energy and to find the correlation matrix of the ground state one has to elucidate all the constraints that a two-point reduced matrix should satisfy. This problem and the whole program are known as the Coulson challenge5[6]. In the form just described it may be unfeasible even for quantum computers [23]. For other approaches and the current state of the art see [26]. This problem is far beyond the scope of our paper. Nevertheless, the characterization of one point reduced matrices given below imposes also new constraints on the higher reduced states.
3.1.1. Constraints on spin and orbital occupation numbers Let’s return to a system of N fermions, this time of smallest possible spin s = 1/2, dim Hs = 2. In this case
spin-orbital decomposition (17) involves only terms Hνr ⊗ Hν
t
s (22)
with at most a two-column diagramν. The sizes of the columns α ≥ β are determined by equations
α + β = N, α − β = 2J, (23)
4 More precisely, the occupation numbers of natural orbitals. The latter are defined as eigenvectors of the
particle density matrix.
5 Also known as two-particle N -representability or, following D. Herschbach, a holy grail of theoretical
where J is the total spin of the system, so thatHνst = HJis just the spin J representation
of the group SU(Hs) = SU(2).
Consider now a pure N -fermion state of total spin J ψ ∈ Hrν⊗ HJ,
where the diagramν is determined by Eqs. (23). Letρν andρJ be its reduced states in the orbital and spin components respectively. The basic fact is that the reduced states are isospectral Specρν = Spec ρJ. Hence Specρνcan be identified with the spin occupation numbers. On the other hand Theorem1, in view of Proposition1, relates Specρνwith the orbital occupation numbers given by the spectrum of the particle density matrix Nρ. In this way one can produce all constraints on allowed spin and orbital occupation numbers, provided that a solution of theν-representability problem is known for two-column diagrams. We address this issue in Sects.3.2and3.3. See also Corollary1in Sect.3.2.
3.2. Formal solution of theν-representability problem. Henceforth we treat the lower index r as the rank of the Hilbert spaceHr. Recall that the character of the representation
Hν
r, i.e. the trace of a diagonal operator
z= diag(z1, z2, . . . , zr) ∈ U(Hr), (24)
in some orthonormal basis e ofHr, is given by Schur’s function Sν(z1, z2, . . . , zr). It
has a purely combinatorial description in terms of the so called semistandard tableaux T of shapeν. The latter are obtained from the diagram ν by filling it with numbers 1, 2, . . . , r strictly increasing in columns and weakly in rows. Then the Schur function can be written as a sum of monomials zT =
i∈Tzi,
Sν(z) =
T
zT,
corresponding to all semistandard tableaux T of shapeν. The monomials are actually the weights of representationHνr, meaning that
z· eT = zTeT (25)
for some basis eT ofHνr parameterized by the semistandard tableaux. Denote by t⊂
u(Hr) and tν ⊂ u(Hνr) the Cartan subalgebras of real diagonal operators in the bases e
and eT respectively, so that the differential of the above group action z : eT → zTeT
gives the morphism
f∗: t → tν, f∗(a) : eT → aTeT, (26)
where aT := i∈Tai. As in Example 2 we treat the orbits Oa and Of∗(a) as flag
varietiesFa(Hr) and Faν(Hνr) consisting of Hermitian operators of spectra a : a1 ≥
a2≥ · · · ≥ ar and aνrespectively. Here aνconsists of the quantities aT arranged in the
non-increasing order
aν := {aT | T = semistandard tableau of shape ν}↓. (27)
Finally, we need the morphism
together with its cohomological version
ϕa∗: H∗(Faν(Hrν)) → H∗(Fa(Hr)), (29)
given in the canonical bases by coefficients cvw(a): ϕa∗: σw →
v
cvw(a)σv. (30)
Theorem 2. In the above notations all constraints on the occupation numbersλ of the
systemHrνin a stateρνof spectrumµ are given by the inequalities
i aiλv(i)≤ k aνkµw(k) (31)
for all test spectra a and permutationsv, w such that cvw(a) = 0.
Proof. In view of Proposition1, this is what Theorem1tells. One has to remember that the left action of a permutation on “places” is inverse to its right action on indices. That is why the permutationsv and w, acting on a and f∗(a) = aν in Theorem1, move to the indices ofλ and µ in the inequality (31).
The coefficient cwv(a) depends only on the order in which quantities aT appear in the
spectrum aν. The order changes when the test spectrum a crosses a hyperplane HT|T : i∈T ai = j∈T aj.
The hyperplanes cut the set of all test spectra into a finite number of polyhedral cones called cubicles. For each cubicle one has to check the inequality (31) only for its extremal edges. As a result, theν-representability amounts to a finite system of linear inequalities. Remark 2. Let’s emphasize once again the difference between the Berenstein-Sjamaar Theorem [1, Thm 3.2.1] and its version used in this paper. In the settings of Theorem2
it manifests itself in the way the quantities aT are ordered in the spectrum aν, or which
parabolic subgroup is used for definition of Schubert cocycles. Berenstein and Sjamaar choose a specific order of tableaux T , while we rely on the natural order of the quantities aT =
i∈Tai. The latter choice allows to treat the inequalities uniformly, and to avoid a
rather cumbersome transformation every time the test spectrum passes from one cubicle to another.
Recall from Sect. 3.1.1that the theorem also describes a relationship between the spin and orbital occupation numbers. We keep for them the above notationsµ and λ respectively.
Corollary 1. All constraints on spin and orbital occupation numbers of N -electron
system in a pure state of total spin J are given by the inequalities (31), applied to two column diagramν determined by Eqs. (23), and bounded to mixed statesρνof rank not exceeding dimensionality 2 J + 1 of the spin space.
We postpone the calculation of the coefficients cwv(a) to Sect.3.3and focus instead on some general results that can be deduced from the theorem as it stands.
3.2.1. Basic inequalities Being a ring homomorphism,ϕa∗maps unit into unitϕ∗a(1) = 1, that is cvw(a) = 1 for identical permutations v, w. Hence the following basic inequality
i aiλi ≤ k akνµk
holds for all test spectra a. Let’s look at it more closely for a pure stateρν = |ψψ| in which case the right-hand side is maximal and the inequality takes the form
i aiλi ≤ a1ν = max T i∈T ai = i aiνi, (32)
whereν1≥ ν2≥ · · · ≥ 0 are rows of ν. The maximum in the right-hand side is attained
for the tableau T of shapeν whose i-row is filled by i. The normalizationiλi = N =
jνjallows to shift the test spectra into the
posi-tive domain a1≥ a2≥ · · · ≥ 0, so that they become nonnegative linear combinations
of the fundamental weights
ωk= (1, 1, . . . , 1 k
, 0, 0, . . . , 0). (33)
Hence it is enough to check (32) for a = ωk, that gives the majorization inequality
λ ν, cf. Example 1. Thus we arrive at the first claim of the following result that characterizes occupation numbers of systemHνin an unspecified mixed state.
Theorem 3. The occupation numbers of the systemHνin an arbitrary mixed state satisfy the majorization inequality
λ ν, (34)
and any suchλ can be realized as the occupation numbers of some mixed state. Proof. The second claim follows from two observations:
1. The occupation numbers of a coherent stateψ ∈ Hν, that is a highest vector of the representation, are equal toν.
2. The set of allowed occupation numbers, written in any order, form a convex set. Indeed, the polytope given by the majorization inequality (34) is just a convex hull of vectors obtained fromν by permutations of coordinates, cf. Example1. Hence by 1 and 2 it consists of legitimate occupation numbers.
Proof of 1. Consider a decomposition of the complexified Lie algebra
u(H) ⊗ C = gl(H) = n−+ h + n+,
into a diagonal Cartan subalgebra h = t ⊗ C accompanied with lower- and upper-triangular nilpotent subalgebras n∓. By definition n+ annihilates the highest vector
ψ ∈ Hν of weight ν. Hence ψ|X±|ψ = X∓ψ|ψ = 0 for all X± ∈ n±. Then by Eq. (19)
ψ|X±|ψ = Tr
Hν(X±|ψψ|) = TrH(X±f∗(|ψψ|)) = 0, ∀X±∈ n±. This means thatρ = f∗(|ψψ|) is a diagonal matrix. On the other hand tψ = t, νψ for t ∈ t, hence as above
t, ν = ψ|t|ψ = TrHν(t|ψψ|) = TrH(t f∗(|ψψ|)) = TrH(tρ) = t, ρ,
Proof of 2. Letρ1ν,ρ2νbe mixed states, with the particle densitiesρ1,ρ2, and the
occupa-tion numbersλ1,λ2. We apply toρ1,ρ1νa unitary rotationρ1 → Uρ1U∗,ρ1ν → Uρ1νU∗
that transforms orthonormal eigenvectors ofρ1into that ofρ2in a prescribed order. The
resulting new operatorsρ1,ρ2commute and have the original spectraλ1, λ2. Then the
particle density matrixρ = p1ρ1+ p2ρ2of the convex combinationρν = p1ρ1ν+ p2ρ2ν
has spectrumλ = p1λ1+ p2λ2.
For a column diagram ν the majorization inequality λ ν amounts to the Pauli exclusion principleλi ≤ 1. In general, we refer to it as the Pauli constraint. Note that
the above proof shows that equality in (34) is attained for coherent states only. The second part of Theorem3extends Coleman’s result [5] for∧NH.
Recall, that the theorem solves theν-representability problem for unspecified mixed states. We will see later that for pure states the answer in general is much more com-plicated. Nevertheless, there are surprisingly many systems for which the majorization inequality alone is sufficient for pureν-representability. We address them in the next item.
3.2.2. Pure moment polytope One of the most striking features of Theorem2 is the linearity of the constraints (31). As a result, the allowed spectra(λ, µ) form a convex polytope, called (noncommutative) moment polytope. The convexity still holds for any fixedµ = Spec ρν, and in particular for the occupation numbersλ of all pure states. We refer to the latter case as the pure moment polytope. It sits inside the positive Weyl chamber, and its multiple kaleidoscopic reflections in the walls of the chamber generally form a nonconvex rosette, consisting of all legitimate occupation numbers written in an arbitrary order. It can be convex only if all constraints on the occupation numbers are given by the majorization inequality λ ν alone. Here we describe a class of representationsHν with this property.
This happens, for example, for a system of N ≥ 2 bosons. In this case ν is a row diagram and the majorization inequality imposes no constraints onλ. By Theorem3this means that every nonnegative spectrumλ of trace N represents occupation numbers of some mixed state. However for bosons one can easily find a pure state that does the job:
ψ =
i
λiei⊗N ∈ SNH,
where ei is an orthonormal basis ofH. This makes the bosonic N-representability
pro-blem trivial.
A more interesting physical example constitutes the so-called closed shell, meaning a system of electrons of total spin zero. The corresponding diagramν consists of two columns of equal length. We will see shortly that in this case the Pauli constraintλ ≤ 2 shapes the pure moment polytope.
Observe that it is enough to construct pure states whose occupation numbers are generators of the cone cut out of the Weyl chamber by the majorization inequality λ ν. Then the convexity does the rest.
Recall that in the proof of Theorem3we have already identifiedν with the occu-pation numbers of a coherent state. Due to the majorization inequality λ ν, the entropy of its reduced state is minimal possible. For that reason coherent states are generally considered closest to classical ones [30]. At the other extreme one finds the so-called completely entangled statesψ ∈ Hν whose particle density matrixρ = f∗(|ψψ|) is scalar and the reduced entropy is maximal [19]. By definition (19) we have
TrH(Xρ) = TrHν(X|ψψ|) = ψ|X|ψ, so that the completely entangled states can
be described by the equation
ψ|X|ψ = 0, ∀ X ∈ su(H). (35)
Let’s call a systemHνr exceptional if the SU(Hr)-representation Hνr is equivalent to one
of the following:Hr, its dualHr∗, and, for odd rank r ,∧2Hr,∧2H∗r. The Young diagram
ν of an exceptional system can be obtained from an r × m rectangle by adding an extra column of length 1, r − 1, 2, r − 2 respectively.
One readily realizes that the exceptional systems contain no completely entangled states, say because the reduced matrix ofψ ∈ ∧2H
r has an even rank.
Proposition 2. In every non-exceptional systemHνthere exists a completely entangled state.
Proof. The result is actually well known, but in a different context. The entanglement equation (35) is nothing but the stationarity condition for the length of vectorψ|ψ with respect to the action of the complexified group SL(H). It is known [34] that every stationary point is actually a minimum, and an SL(H)-orbit contains a minimal vector if and only if the orbit is closed. As a result, we end up with the problem of existence of a nonzero closed orbit, or, what is the same, the existence of a nonconstant polynomial invariant. The proposition just reproduces a known answer to the latter question [34].
By admitting other simple Lie groups we find only two more exceptional repre-sentations: the standard representation of the symplectic group Sp(n) and a halfspinor representation of Spin(10).
Now we can solve the pureν-representability problem for a wide class of systems, including the above mentioned closed shell.
Theorem 4. Suppose that all columns of Young diagramν are multiple, meaning that
every number in the sequence of column lengthsν1t ≥ νt2≥ ν3t ≥ · · · appears at least twice. Then all constraints on the occupation numbers of the systemHνin a pure state are given by the majorization inequalityλ ν alone.
Proof. We’ll proceed by induction on the height of the diagramν. The triviality of the bosonic N -representability problem provides a starting point for the induction.
Let nowλ be a vertex of the polytope cut out of the positive Weyl chamber by the majorization inequalityλ ν. Note that the latter includes the equation Tr λ = Tr ν. Then the following alternative holds:
1. Either all nonzero components ofλ are equal,
2. Or one can splitλ and ν into two parts λ = λ|λ,ν = ν|ν containing the first p components and the remaining ones, both satisfying the inequalitiesλ ν, λ ν.
Indeed, the second claim states that the pthmajorization inequality in (5) turns into an equation. On the other hand, if all the majorization inequalities are strict, andλ contains different nonzero entries, then one can linearly vary these entries preserving the non-increasing order ofλ and the majorization λ ν. As result we get a line segment in the polytope containingλ, which is impossible for a vertex.
We have to prove that every vertexλ represents occupation numbers of some pure state. Consider the above two cases separately.
Case 1. Letλ contain r equal nonzero entries and Hr ⊂ H be a subspace of dimension
r . The conditions of the theorem ensure that the systemHνr is non-exceptional, hence by Proposition2it contains a stateψ ∈ Hrνwith occupation numbers equal to the nonzero
part ofλ. In the bigger system Hν ⊃ Hνr its occupation numbers will be extended by
zeros.
Case 2. Let the system have rank r = p + q. Choose a decomposition Hr = Hp⊕ Hq
and consider a restriction of the representationHνr onto subgroup U(Hp) × U(Hq)
Hrν=
µ,π
cνµπHµp⊗ Hπq, (36)
where cνµπ are the omnipresent Littlewood-Richardson coefficients. Observe that cννν = 1, and therefore Hνp ⊗ Hν
q ⊂ Hrν. By the induction hypothesis there exist
statesψ∈ Hνpandψ∈ Hqνwith occupation numbersλ,λand particle densitiesρ, ρrespectively. Then decomposable stateψ = ψ⊗ ψ has particle densityρ⊕ ρ, and its occupation numbers are equal toλ = λ|λ.
Let’s extract for reference a useful corollary from the last part of the proof.
Corollary 2. Suppose that the Littlewood-Richardson coefficient cνµπis nonzero. Then merging the occupation numbersλ,λof the systemsHµp,Hπq form legitimate
occupa-tion numbers of the systemHνp+q.
Remark 3. The restriction on the column’s multiplicities of diagramν is needed only to ensure that the components of any splittingν = ν|ν|ν| . . . are non-exceptional. The latter condition holds for any two-row diagram[α, β], β = 1 for dim H ≥ 3. This gives examples of systems beyond Theorem4, say forν = [3, 2], whose pure moment polytope is given by the majorization inequality alone. More such diagrams can be produced as follows: takeν as in Theorem4and remove one cell from its last row. This works when the last row contains at least three cells and the rank of the system is bigger than the height ofν. A complete classification of all such systems is still missing. 3.2.3. Dadok-Kac construction In the last two theorems we encounter the problem of constructing a pure state with given occupation numbers. The problem lies at the very heart of theν-representability and one shouldn’t expect an easy solution. Nevertheless, there is a combinatorial construction that produces a state with diagonal density matrix, whose spectrum can be easily controlled. It has been used first by Borland and Dennis [3] to forecast the structure of the moment polytope for small fermionic systems. Later on Müller [27] formalized and advanced their approach to the limit. It fits into a general Dadok-Kac construction [10] that works for any representation.
Below we follow the notations introduced at the beginning of Sect. 3.2. Let x = diag(x1, x2, . . . , xr) be a typical element from the Cartan subalgebra t ⊂ u(Hr).
For a given semi-standard tableau T call the linear formωT : x → xT = i∈Txi
the weight of the basic vector eT ∈ Hνr. We also need nonzero weights of the adjoint
representationαi j : x → xi− xj, i = j called roots. Let’s turn the set of semi-standard
tableaux of shapeν into a graph by connecting T and Teach timeωT − ωT is a root,
Proposition 3. Let T be a set of semi-standard tableaux of shapeν containing no
connec-ted pairs. Then every state ψ = T∈TcTeT ∈ Hν with support T has a diagonal
particle density matrix with entries λi =
Ti
|cT|2, (37)
where every tableau T is counted as many times as the index i appears in it.
Proof. The proof refines the arguments used in Claim 1 of Theorem3, from which we borrow the notation. As in the above theorem we have to proveψ|X|ψ = 0 for every X∈ n++ n−. It is enough to consider root vectors Xαthat form a basis of n++ n−. Then
ψ|Xα|ψ =
T,T∈T
cTcTeT|Xα|eT.
Since XαeT has weightα + ωT, it is orthogonal to eT, except forωT = ωT +α. The
latter is impossible for T, T ∈ T, and therefore the reduced state of ψ is diagonal. A straightforward calculation gives the diagonal entries (37).
We’ll have a chance to use this construction in Sect.4.1.
Note that for a fixed support T the set of unordered spectra (37) form a convex polytope. It is not known when this approach exhausts the whole moment polytope. The smallest fermionic system where it fails is∧3H8, see Sect.6.
3.3. Calculation of the coefficients cvw(a). To progress further and to give Theorem1
full strength one has to calculate the coefficients cvw(a). Berenstein and Sjamaar left this problem mostly untouched. However, in theν-representability settings, highlighted in Theorem2, this can be done very explicitly.
3.3.1. Canonical generators To proceed we first need an alternative description of the cohomology of the flag varietyFa(Hr) [2]. Recall that the latter is understood here as
the set of Hermitian operators inHr of given spectrum a. To avoid technicalities, we
assume the spectrum to be simple, a1> a2> · · · > ar. LetEi be the eigenbundle on
Fa(Hr) whose fiber at X ∈ Fa(Hr) is the eigenspace of operator X with eigenvalue
ai. Their Chern classes xi = c1(Ei) generate the cohomology ring H∗(Fa(Hr)) and we
refer to them as the canonical generators. The elementary symmetric functionsσi(x) of
the canonical generators are the characteristic classes of the trivial bundleHr and thus
vanish. This identifies the cohomology with the ring of coinvariants
H∗(Fa(Hr)) = Z[x1, x2, . . . , xr]/(σ1, σ2, . . . , σr). (38)
This approach to the cohomology is more functorial and for that reason leads to an easy calculation of the morphism (29),
ϕa∗: H∗(Faν(Hν)) → H∗(Fa(H)).
Recall that the spectrum aν consists of the quantities aT =
i∈Tai arranged in
decreasing order, where T runs over all semi-standard tableaux of shapeν. We define xT =
Proposition 4. Let xiand xkνbe the canonical generators of H∗(Fa(H)) and H∗(Faν(Hν))
respectively. Then
ϕa∗(xkν) = xT, when akν = aT. (39)
In other words,ϕ∗a(xkν) is obtained from aνk by the substitution ai → xi.
Proof. The eigenbundleEiis equivariant with respect to the adjoint action X → uXu∗of
the unitary group U(H). Therefore it is uniquely determined by the linear representation of the centralizer D= Z(X) in a fixed fiber Ei(X) or by its character εi : D → S1= {z ∈
C∗| |z| = 1}. In the eigenbasis e of the operator X the centralizer becomes a diagonal torus with typical element z = diag(z1, z2, . . . , zr) and the character εi : z → zi.
Let now Xν = ϕa(X), Dν = Z(Xν), and eT be the weight basis ofHν, introduced
in Sect.3.2, parameterized by the semi-standard tableaux T of shapeν and arranged in the order of eigenvalues aν. Then the character of the pull-backϕa−1(Ekν) is just the weighti∈Tεiof the kthvector eT, where the tableau T is determined from the equation
aνk = aT, cf. (25). Thusϕ−1a (Ekν) =
i∈TEi, and we finally get
ϕa∗(xkν) = ϕa∗(c1(Ekν)) = c1(ϕ−1a (Ekν)) = c1( i∈T Ei) = i∈T xi = xT.
Remark 4. Formula (39) may look ambiguous for a degenerate spectrum a, while in fact it is perfectly self-consistent. Indeed, consider a small perturbation˜a, resolving multiple components of a, and the natural projection
π : F˜a(H) → Fa(H)
that maps X =i ˜ai|eiei| into X =iai|eiei|, where eiis an orthonormal
eigen-basis of X . It is known [2] thatπ induces the isomorphism π∗: H∗(F
a(H)) H∗(F˜a(H))W(D), (40)
where the right-hand side denotes the algebra of invariants with respect to permutations of the canonical generators ˜xi with the same unperturbed eigenvalue ai = α. Such
permutations form the Weyl group W(D) of the maximal torus D= Z( X) in D = Z(X). For example, characteristic classes of the eigenbundleEα with the multiple eigenvalue α = aicorrespond to elementary symmetric functions of the respective variables ˜xi.
Equation (39), as it stands, depends on a specific ordering of the unresolved spectral values ai and akν. However, whenϕa∗is applied to invariant elements with respect to the
above Weyl group, the ambiguity vanishes.
Note also that the Schubert cocycleσw ∈ H∗(F˜a(H)) is invariant with respect to W(D) if and only if w is the shortest representative in its left coset modulo W(D). Such cocycles form the canonical basis of cohomology H∗(Fa(H)).
3.3.2. Schubert polynomials To calculate the coefficients cvw(a) we have to return to the Schubert cocyclesσwand express them via the canonical generators xi. This can be
accomplished by the divided difference operators ∂i : f (x1, x2, . . . , xn) →
f(. . . , xi, xi +1, . . .) − f (. . . , xi +1, xi, . . .)
xi− xi +1
(41) as follows. Write a permutationw ∈ Snas a product of the minimal number of
transpo-sitions si = (i, i + 1),
w = si1si2· · · si . (42)
The number of factors (w) = #{i < j | w(i) > w( j)} is called the length of the permutationw. The product
∂w := ∂i1∂i2· · · ∂i
is independent of the reduced decomposition and in terms of these operators the Schubert cocycleσwis given by the equation
σw= ∂w−1w0(x1n−1x2n−2· · · xn−1), (43)
wherew0= (n, n − 1, . . . , 2, 1) is the unique permutation of maximal length.
The right-hand side of Eq. (43) makes sense for independent variables xi and in this
setting it is called the Schubert polynomial Sw(x1, x2, . . . , xn), deg Sw = (w). They
were first introduced by Lascoux and Schützenberger [21,22] who studied them in a series of papers. See [24] for further references and a concise exposition of the theory. We borrow from [21] the following table, in which x, y, z stand for x1, x2, x3:
w Sw w Sw w Sw w Sw 3210 x3y2z 2301 x2y2 2031 x2y + x2z 1203 x y 2310 x2y2z 3021 x3y + x3z 2103 x2y 2013 x2 3120 x3yz 3102 x3y 3012 x3 0132 x + y + z 3201 x3y2 1230 x yz 0231 x y + yz + zx 0213 x + y 1320 x2yz + x y2z 0321 x2y + x2z + x y2 0312 x2+ x y + y2 1023 x 2130 x2yz 1302 x2y + x y2 1032 x2+ x y + x z 0123 1
Extra variables xn+1, xn+2, . . . added to (43) leave the Schubert polynomials
unal-tered. For that reason they are usually treated as polynomials in an infinite ordered alphabet x = (x1, x2, . . .). With this understanding every homogeneous polynomial can
be decomposed into Schubert components as follows:
f(x) =
(w)=deg( f )
∂wf · Sw(x). Applying this to the polynomial
ϕa∗(Sw(xν)) = Sw(ϕa∗(xν)) =
(v)= (w)
cvw(a) · Sv(x),
Theorem 5. For the ν-representability problem the coefficients of the decomposition
ϕ∗
a(σw) =
vcvw(a)σvare given by the formula
cwv(a) = ∂vSw(xν) |xνk →xT, (44)
where the tableau T is derived from equation aνk = aT, and the operator∂vacts on the
variables xi, replacing xkν via specialization xkν → xT =
i∈Txi.
Note that this equation is independent of an ordering of the unresolved spectral values aνk. Indeed, the Schubert polynomial Sw(xν) is symmetric in the respective variables xkν, provided thatw is the minimal representative in its left coset modulo the centralizer of the spectrum aν in the symmetric group. Only such permutations correspond to the Schubert cocyclesσw ∈ H∗(Faν(Hν)), cf. Remark4.
4. Beyond the Basic Constraints
Here we use the above results to derive some general inequalities for the pure ν-representability problem beyond the Pauli constraint λ ν. We start with a complete solution of the problem for two-row diagrams, and then turn to the initial N -representability problem that appears to be the most difficult one.
4.1. Two-row diagrams. For the two-row diagramν = [α, β] the majorization inequa-lity λ ν just tells us that λ1 ≤ α. As we know, for β = 1 it shapes the whole
moment polytope, see Remark3to Theorem4. Here we elucidate the remaining case ν = [N − 1, 1], and thus solve the pure ν-representability problem for all two-row diagrams. The result can not be extended to three-row diagrams, nor even to three fer-mion systems, where the number of independent inequalities increases with the rank, see Corollary3below. For convenience and future reference we collect all known facts in the next theorem.
Theorem 6. For a system Hrν of rank r ≥ 3 with the two-row diagram ν = [α, β],
α + β = N, all constraints on the occupation numbers of a pure state are given by the following conditions:
1. Basic inequality:λ1≤ α for β = 1.
2. Inequality:λ1− λ2≤ N − 2 for ν = [N − 1, 1], N > 3.
3. Inequalities:λ1− λ2≤ 1, λ2− λ3≤ 1 for ν = [2, 1].
4. Even degeneracy:λ2i−1= λ2iforν = [1, 1].
Proof. We have already addressed Cases 1 and 4 in Remark 3 and the Introduction respectively.
Case 2. Necessity. To prove the inequalityλ1− λ2≤ N − 2 we have to put it into the
form of Theorem2: i aiλv(i)≤ k aνkµw(k). (45)
This suggests the test spectrum a= (1, 0, 0, . . . , 0, −1) and the shortest permutation v that transforms it into(1, −1, 0, 0, . . . , 0), which is the cyclic one v = (2, 3, 4, . . . , r).
Thus we get the left-hand side of the inequality. To interpret its right-hand side N − 2, notice that the spectrum aν starts with the terms
aν = (N − 1, N − 1, . . . , N − 1
r−2
, N − 2, . . .),
corresponding to semi-standard tableaux T with first row of ones and the indices 2, 3, . . . , r filling a unique place in the second row. Since for pure state µ = (1, 0, 0, . . . , 0), then the shortest permutationw that produces N − 2 in the right-hand side of (45) is also cyclic,w = (1, 2, 3, . . . , r − 1). The corresponding Schubert polynomial is just the monomial
Sw(xν) = x1νx2ν· · · xrν−2.
This is a special case of Grassmann permutations discussed in the next Sect. 4.2. Specialization xkν → xT of Theorem5transforms it into the product
P(x) =
r−1 i=2
[(N − 1)x1+ xi].
Taking the reduced decompositionv = s2s3· · · sr−1we infer
cvw(a) = ∂vP(x) = ∂2∂3· · · ∂r−1P(x).
The right-hand side is a constant, and the operators∂i do not touch x1. Hence we can
put x1= 0, that gives
cwv(a) = ∂2∂3· · · ∂r−1(x2x3· · · xr−1) = 1.
Since cvw(a) = 0, the inequality follows from Theorem2.
Case 2. Sufficiency. By the convexity it is enough to construct extremal states whose occupation numbers are vertices of the polytope cut out of the Weyl chamber by the inequalityλ1− λ2 ≤ N − 2 and the normalization Tr λ = N. The vertices are given
first of all by the fundamental weights normalized to trace N ωk = (N/k, N/k, . . . , N/k
k
, 0, 0, . . . , 0)
that generate the edges of the Weyl chamber, except forω1forbidden by the constraint
λ1− λ2 ≤ N − 2. The latter is replaced by the intersections τk of segments[ω1, ωk]
with the hyperplaneλ1− λ2= N − 2,
τk = (N − 2 + 2/k, 2/k, . . . , 2/k k
, 0, 0, . . . , 0).
Here we tacitly assume that N > 3, since otherwise ω2would be also forbidden. The
same condition ensures that the systemHkνis non-exceptional for k ≥ 2, hence ωk are
occupation numbers of some pure states by Proposition2.
To deal with the remaining vertices τk we invoke the Dadok-Kac construction,
Sect.3.2.3and observe that the state ψk= k1 k k · · · k + 1 √ 2 2≤i<k i i k · · · k k
has disconnected support and the occupation numbersτk, k ≥ 2. Here for clarity we
write tableau T instead of the weight vector eT and skip an overall normalization factor.
Case 3. Here we only briefly sketch the proof which follows a similar scheme. The second inequality in the formλ2−λ3≤ N −2 holds for all N, but it becomes redundant
for N > 3. It can be deduced from Theorem2by calculation of the coefficient cvw(a) for the same a andw as above, but with another permutation v = (1, 2)(3, 4, . . . , r). Then, keeping the notations of Case 2, we get
cvw(a) = ∂3∂4· · · ∂r−1∂1P(x1, x2, . . . , xr−1)
= ∂3∂4· · · ∂r−1
P(x1, x2, . . . , xr−1) − P(x2, x1, . . . , xr−1)
x1− x2 .
The operators ∂k, k ≥ 3 do not affect the variables x1, x2. Therefore in the fraction
we can pass to the limit x1, x2 → 0 equal to (N − 2)x3x4· · · xr−1, which gives
cvw(a) = N − 2 = 0.
To prove sufficiency of the above inequalities we again have to look at the vertices of a polytope cut out of the Weyl chamber by the constraintsλ1− λ2≤ 1, λ2− λ3≤ 1,
Trλ = 3. This time, along with ωk, k ≥ 3 and τk, k ≥ 2, there are vertices of another
type
ηk= (1 + 1/k, 1 + 1/k, 1/k, 1/k, . . . , 1/k k
, 0, 0, . . . , 0)
for k ≥ 3. They represent occupation numbers of the following states with disconnected support: ψk= √ k + 1 1 1 2 + √ 2 2 2 3 + 3<i≤k 2 i i .
Remark 5. Two-row diagrams naturally appear in the description of bosonic systems, like photons where polarization plays the rôle of spin. Representation with diagram can be applied both for bosons and fermions. In this case we calculated all constraints on the spin and orbital occupation numbers for small ranks, see Sect.6.1. It appears that the constraints are stable and independent of the rank.
4.2. Grassmann inequalities. Let’s return to the initial pure N -representability problem for system∧NHr and consider a constraint on its occupation numbers with 0/1
coeffi-cients,
λi1+λi2+· · · + λip ≤ b, (46)
called the Grassmann inequality. For example, all constraints (4) for the system∧3H7
are Grassmannian. We assume that the Grassmann inequality is essential, meaning that it defines a facet of the moment polytope. Then it should fit into the form of Theorem2
with
a= (1, 1, . . . , 1
p
and the Grassmann permutation or shuffle
v = [i1, i2, . . . , ip, j1, j2, . . . , jq] := [I, J], (47)
where I and J are increasing sequences of lengths p and q, p + q = r. This is the shortest permutation that produces the left-hand side of inequality (46). Our terminology stems from the observation that for the test spectrum a the flag varietyFa(H) reduces
to the Grassmannian Grqp(H) consisting of all subspaces in H of dimension p and
codimension q.
It is instructive to think about the Grassmann permutationv = [I, J] geometrically as a path connecting the SW and N E corners of the p × q rectangle, with the kthunit step running to the North for k ∈ I and to the East for k ∈ J. The path cuts out of the rectangle a Young diagramγ at its N W corner. We’ll refer to I and J as the vertical and horizontal sequences of the diagramγ ⊂ p × q and denote the corresponding shuffle byvγ = [I, J]. The length of the shuffle vγ is equal to the size|γ | of the diagram γ and its Schubert polynomial reduces to the much better understood Schur function
Svγ(x) = Sγ(x1, x2, . . . , xp).
Observe thatγp−k+1 = ik − k, and the size of the Young diagram γ is related to its
vertical sequence by the equation
|γ | =
1≤k≤p
(ik− k). (48)
To get the strongest inequality (46) we choosew to be a cyclic6permutation w = (1, 2, . . . , + 1) = [2, 3, . . . , + 1, 1, + 2, + 3 . . . , r] of length = (v) = |γ | for which the right-hand side b = (∧Na)
+1of (45) is minimal and equal to the( + 1)stterm of the non-increasing sequence
∧N
a = {aK := ak1+ ak2 +· · · + akN | 1 ≤ k1< k2< · · · < kN ≤ r}↓.
The sequence consists of nonnegative numbers m, each taken with multiplicity
p m q N− m .
Recall that w also should be the minimal representative in its left coset modulo the stabilizer of∧Na. For the cyclic permutation this amounts to the inequality(∧Na)
> (∧Na)
+1 = b, which tells us that the first terms of ∧Na contain all the components bigger than b. The number of such terms is bounded by the inequality
m>b p m q N − m = = |γ | ≤ pq. (49)
To avoid sporadic constraints, assume that the inequality we are looking for is stable, i.e. remains valid for an arbitrary big rank r . Then the left-hand side should be linear in
q = r − p and the sum contains at most two terms: m = N and m = N − 1. Thus we
end up with two possibilities:
6 Actuallyw is always cyclic for an essential pure ν-representability inequality. We’ll address this issue
1. b= N − 2, p = N − 1, = r − p, that gives the inequality
λi1 +λi2+· · · + λiN−1 ≤ N − 2, (50)
withk(ik− k) = r − p.
2. b= N − 1, p ≥ N, =Np, that gives the inequality
λi1+λi2+· · · + λip ≤ N − 1, (51) withk(ik− k) = p N .
We will refer to them as the Grassmann inequalities of the first and second kind respecti-vely. For the inequalities of the first kind the sumk(ik−k) = r − p increases with the
rank, and therefore some of the involved occupation numbers should move away from the head of the spectrum. In contrast, the constraints of the second kind deal only with a few leading occupation numbers that are independent of the rank. We analyze them below for p= N + 1 and postpone the more peculiar first kind to the next section. The final result is that these inequalities actually hold true with very few exceptions.
The cyclic permutationw is a special type of shuffle with the column Young diagram of height . The corresponding Schur function is just the monomial
Sw(y) = y1y2. . . y .
Applying to Swthe specialization of Theorem5we arrive at the product
P(x) = 1≤k1<k2<···<kN≤p (xk1+ xk2+· · · + xkN) = γ cγSγ(x1, x2, . . . , xp). (52)
Being symmetric, it can be expressed via Schur functions and, by Theorem2, each time Sγ(x) enters into the decomposition with nonzero coefficient cγ = 0 we get inequality
λi1+λi2 +· · · + λip ≤ N − 1, (53)
where i1 < i2 < · · · < ip is the vertical sequence of Young diagramγ ⊂ p × q,
|γ | =p N
.
The product P(x) represents the top Chern class of the exterior power ∧NEpof the
tautological bundleEpon the Grassmannian Gr q
pand the decomposition (52) has been
discussed in this context [20]. However, known results are very limited. Example 3. For N = 2 and any p ≥ N the product
P(x) =
1≤i< j≤p
(xi + xj) = Sδ(x1, x2, . . . , xp)
is just the Schur function with a triangular Young diagramδ = [p − 1, p − 2, . . . , 0], see [25]. This gives for the two fermion system∧2H the inequality
λ1+λ3+λ5+λ7· · · ≤ 1, (54)
that, due to the normalizationiλi = 2, degenerates into equality and implies even the
degeneracyλ2i−1= λ2i of the occupation numbers.
On the other hand, for arbitrary N and minimal value p= N we get P(x) = x1+ x2+· · · + xN = S(x).
